Elliptic curves enter algebraic topology through “Elliptic cohomology”–really a family of cohomology theories–and their associated “elliptic genera”. • Arithmetic aspect: Modularity of elliptic genera, The spectrum TMF of “topological modular forms” and the calculation of π∗TMF →MF (Z), Hopkins’s proof of Borcherds’ congruences. • Physical aspect: Witten’s approach to elliptic genera via string...

The evolutional course of music through centuries has shown an incremental use of chromatic variations by composers and performers for melodies’ and music sounds’ enrichment. This paper presents an integrated model, which contributes to the calculation of musical chromaticism. The model takes into account both horizontal (melody) and vertical chromaticism (harmony). The proposed perceptually ba...

In the first part of these notes we use a probabilistic method to show the existence of graphs with large girth and large chromatic number. In the second part we give an explicit example of such graphs. It is mostly based on the third chapter of Some Applications Of Modular Forms by Peter Sarnak and also the third and forth chapters of Elementary Number Theory, Group Theory, And Ramanujan Graph...

Suppose that G is a graph with maximum degree d(G) and for every vertex v in G, the neighborhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/ log f , for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [Kim], Johansson [Joh], Alon, Krivelevich and Sudakov [AKS...

Photonic Crystal Fibers (PCFs) can be used in optical communications as transmission lines. For this reason, the PCFs with low confinement loss, low chromatic dispersion, and low nonlinear effects are highly suitable transmission media. In this paper, we introduce a new design of index-guiding nanostructured photonic crystal fiber (IG-NPCF) with ultra-low chromatic dispersion, low nonlinearity ...

A proper edge colouring of a graph is said to be acyclic if every cycle of G receives at least three colors. The acyclic chromatic index, denoted , is the least number of colors required for an acyclic edge color of . This paper shows an upper bound of the acyclic chromatic index of a class of graphs which can be expressed as the Cartesian product of some graphs. We also give exact values for s...

An undirected simple graphG is locally irregular if adjacent vertices ofG have different degrees. An edge-colouring φ ofG is locally irregular if each colour class of φ induces a locally irregular subgraph of G. The irregular chromatic index χirr(G) of G is the least number of colours used by a locally irregular edge-colouring of G (if any). We show that the problem of determining the irregular...

Let G be a simple graph of order n, and let ∆(G) and χ′(G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ′(G) = ∆(G) or ∆(G) + 1. Following this result, G is called edge-chromatic critical if χ′(G) = ∆(G) + 1 and χ′(G − e) = ∆(G) for every e ∈ E(G). In 1968, Vizing conjectured that if G is edge-chromatic critical, then the independence number α(G) ≤ n/2, ...

A homomorphism from an oriented graph G to an oriented graph H is a mapping φ from the set of vertices of G to the set of vertices of H such that −−−−−−→ φ(u)φ(v) is an arc in H whenever −→ uv is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (th...

There are five possible structures for a set of three lines of a Steiner triple system. Each of these three-line "configurations" gives rise to a colouring problem in which a partition of all the lines of an STS( v) is sought, the components of the partition each having the property of not containing any copy of the configuration in question. For a three-line configuration B, and STS ( v) S, th...